## Cantor Diagonal Argument disproof

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### Re: Cantor Diagonal Argument disproof

All mathematics is, is making up a set of rules, and making logical deductions from them. Euclid started it.robert 46 wrote:Well, if you make up your own rules you can prove whatever you want. That is generally how religions do it.

Your problem is that you refuse to recognize that there can be more than one set of rules, and that the ones you seem to think apply to "all mathematics" have a logical flaw: either endless sets exist, or a largest natural number must exist. And a property of an endless set like N, is that you can match a subset, say E, with N itself. To have the matching process end in a mismatch would require one of the sets to end. And you know this, which is why you ignore this fact by reverting to calling it a fantasy.

None have any connection to reality. Perfect points, lines, circles, and even numbers like "one" don't exist. Some are just closer models of reality than others.The fault is in axioms which have no connection with reality.

No. You are the one claiming it needs to be. All you really need is a definition of every member, AND WE HAVE THAT. Without even building it on the previous ones, which is the definition of "algorithmic" that you refuse to provide. (And, btw, not the one Gofer uses. I never said that Cantor's method was not a way to define a mapping, just that that is not robert's definition of "algorithmic.")Claiming that an infinity of characters can be selected and negated all in parallel is algorithmic?

You are right, I didn't compare you and them. It was a condemnation of your discrimination techniques.It is not a comparison between me and them, ...

- JeffJo
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### Re: Cantor Diagonal Argument disproof

JeffJo wrote:All mathematics is, is making up a set of rules, and making logical deductions from them. Euclid started it.robert 46 wrote:Well, if you make up your own rules you can prove whatever you want. That is generally how religions do it.

Your problem is that you refuse to recognize that there can be more than one set of rules, and that the ones you seem to think apply to "all mathematics" have a logical flaw: either endless sets exist, or a largest natural number must exist. And a property of an endless set like N, is that you can match a subset, say E, with N itself. To have the matching process end in a mismatch would require one of the sets to end. And you know this, which is why you ignore this fact by reverting to calling it a fantasy.

There is a middle way between the closed finite set and the closed infinite set: the open-ended finite set. This naturally embodies the concept of endlessness. For as far as you want to go you can match E with N.

None have any connection to reality. Perfect points, lines, circles, and even numbers like "one" don't exist. Some are just closer models of reality than others.The fault is in axioms which have no connection with reality.

Closed infinite sets are not any kind of model of reality. Open finite sets are.

No. You are the one claiming it needs to be. All you really need is a definition of every member, AND WE HAVE THAT. Without even building it on the previous ones, which is the definition of "algorithmic" that you refuse to provide.Claiming that an infinity of characters can be selected and negated all in parallel is algorithmic?

As I said, if one can make the rules one can prove whatever one wants.

You are right, I didn't compare you and them. It was a condemnation of your discrimination techniques.It is not a comparison between me and them,...[ but between rational thought and irrational thought. I have pointed out the religious leanings of Cantor, and how he injected them into mathematics through the religious method of exaggeration. ]

"Discrimination: recognition and understanding of the difference between one thing and another." [1]

"Discriminatory: making or showing an unfair or prejudicial distinction between different categories of people or things..."

Words change meaning over time, and ignorant group-think has tried to make "discrimination" and "discriminatory" synonyms.

*****

[1]

Discrimination: Late Latin discriminatio, act of contrasting opposite thoughts, separation, distribution, from Latin discriminatus

1a: the act or an instance of discrimination: as

1: the making or perceiving of a distinction or difference <incapable of ~ between the imaginary and the real> <the same name was applied to both instruments with little ~>

2: recognition, perception, or identification esp. of differences <the eye is capable of much finer ~ of detail>

: critical evaluation or judgment <the public would need to be educated in the ~ of cider>

b psychol: the process by which two stimuli differing in some aspect are responded to differently: differentiation

2 archaic: something that discriminates: a distinguishing mark

3: the quality of being discriminating: the power of finely distinguishing (as in respect to quality) : good or refined taste: discernment <nobody should reproach them for reading indiscriminately... only by so doing can they learn ~>

- Webster's 3rd New International Dictionary

Only thereafter does the dictionary cover prejudicial meanings.

Discriminatory: discriminative

esp: applying or favoring discrimination in treatment <a ~ tax> <~ attitudes toward minority groups>

- ibid

Last edited by robert 46 on Sun May 28, 2017 9:45 am, edited 2 times in total.

- robert 46
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### Re: Cantor Diagonal Argument disproof

And you have not defined this concept, nor shown that it is different than Cantor's.robert 46 wrote:There is a middle way between the closed finite set and the closed infinite set: the open-ended finite set. This naturally embodies the concept of endlessness. For as far as you want to go you can match E with N.

Never said they were. Dimensionless points are also not any kind of model of reality, you just accept them more readily.Closed infinite sets are not any kind of model of reality. Open finite sets are.

And as I said, and you (typically) ignored, that is what Mathematics is. Cantor starts with a consistent set of rules, and derives consistent properties from them. Mathematics. And like any Mathematics, they do not represent reality exactly, because that would be the true fantasy. And as I have told you for over a year, and you have (typically) ignored, Cantor's conclusions apply only within his rules. If you want to reject the Axiom of Infinity, that is your prerogative. But you can't "disprove" Cantor's theorem by doing so.As I said, if one can make the rules one can prove whatever one wants.

You, on the other hand, have not defined, or used, a consistent set of rules. Because you known that any attempt to do so will be inconsistent. For one thing, it can't tell you if their are more natural numbers than even natural numbers.

Yes, they do change. But not like you imply. Here's what you probably skipped in your unnamed source:"Discrimination: recognition and understanding of the difference between one thing and another."

"Discriminatory: making or showing an unfair or prejudicial distinction between different categories of people or things..."

Words change meaning over time, ...

"Discrimination: (1) the unjust or prejudicial treatment of different categories of people or things, especially on the grounds of race, age, or sex. (2) recognition and understanding of the difference between one thing and another."

"Discriminatory" describes a single act. The first definition of "Discrimination" in modern language refers to the continuing practice of discriminatory acts. So you just lied, again.

+++++

Example of robert's various ploys for lying:

On Sun May 28, 2017, at 6:49 am, he quoted, without saying from where, this definition: "Discrimination: recognition and understanding of the difference between one thing and another."

At 8:08 am, I found his apparent source with a simple Google source, and quoted it. His meaning was the second definition, and the one he was arguing against was the first. He apparently skipped it, giving the false impression (I.e., a lie) not that his meaning was preferred, BUT THAT IT WAS THE ONLY MEANING.

So robert edited his post twice. I can't say what lies were erased from the first edit, but the second had two. What he implied was the "preferred" meaning does not limit the meaning to what he wants, it also covers perceived differences. But order does not, as robert wants use to believe, indicate preference. It merely discriminates between context. And the context here was the form robert still skipped.

Last edited by JeffJo on Wed Jun 07, 2017 9:36 am, edited 1 time in total.

- JeffJo
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### Re: Cantor Diagonal Argument disproof

robert 46 wrote:Remember, infinite is just a synonym for endless. It is impossible for one endless set to be bigger than another endless set... .

No, it isn't. On the number line, there are more real numbers than there are rational numbers.

There are "holes" on the number line if only the rational numbers are there. When the set of irrational

numbers are included (added), then all of the real numbers are present on the number line, and there

are no "holes," or gaps. The set of real numbers have a greater density than the set of rational numbers.

- phobos rising
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### Re: Cantor Diagonal Argument disproof

which is precisely why one requires precise definitions for "no holes or gaps", "more real numbers than rationals" and "greater density". In this case, that would be the definition of cardinality and "the least-upper-bound property" of the reals.phobos rising wrote: On the number line, there are more real numbers than there are rational numbers. There are "holes" on the number line if only the rational numbers are there. When the set of irrational numbers are included (added), then all of the real numbers are present on the number line, and there are no "holes," or gaps. The set of real numbers have a greater density than the set of rational numbers.

- Gofer
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### Re: Cantor Diagonal Argument disproof

phobos rising wrote:robert 46 wrote:Remember, infinite is just a synonym for endless. It is impossible for one endless set to be bigger than another endless set... .

No, it isn't. On the number line, there are more real numbers than there are rational numbers.

There are "holes" on the number line if only the rational numbers are there. When the set of irrational numbers are included (added), then all of the real numbers are present on the number line, and there are no "holes," or gaps. The set of real numbers have a greater density than the set of rational numbers.

Numbers are discrete entities. The spacetime continuum is considered not to be composed of discrete entities, but is a monolithic whole which is smooth and continuous. Thus we get a fundamental contradiction in talking about a continuum of discrete entities.

If two points on a line are not the same point then there is at least one point between them. Proof: P3=(P1+P2)/2=P1/2+P2/2=P1+P2/2-P1/2=P1+(P2-P1)/2; if P1<P2 then P1<P3<P2. Thus there are always gaps between any two different points. Points are not gap fillers; they are gap makers: i.e. for any gap, a point can be derived which divides the gap into two gaps defined by end points. A line segment, defined by end points, is a gap; but the end points are not part of the gap. Thus a line segment is not composed of points: it is arbitrarily composed of one or more gaps. The gaps are necessarily smooth and continuous.

Simply: a point is a zero-dimensional object; a line segment is a one-dimensional object; it is impossible to make a one-dimensional object out of zero-dimensional objects. A one-dimensional object can only be made from one-dimensional objects. Points have nothing to do with the structure of a line; they are only relevant to the measurement of a line, which is arbitrarily imposed on a line from the outside.

- robert 46
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### Re: Cantor Diagonal Argument disproof

As usual, robert frames his lies by misrepresenting the facts, and criticizing the misrepresentation.robert 46 wrote:Numbers are discrete entities. The spacetime continuum is considered not to be composed of discrete entities ... [for robert: the rest is deleted, since it will be addressed] ...

Simply: a point is a zero-dimensional object; a line segment is a one-dimensional object; it is impossible to make a one-dimensional object out of zero-dimensional objects. A one-dimensional object can only be made from one-dimensional objects. Points have nothing to do with the structure of a line; they are only relevant to the measurement of a line, which is arbitrarily imposed on a line from the outside.

Nobody (who understands math, and does not confuse the set-theory continuum with the space-time continuum) claims that points make up (I.e., provide the structure of) a line, like bricks make up a wall. But points do "lie on" the line. The property that a line segment it is infinitely divisible is what defines the continuum, and provides the extra dimension.

But please tell me: are there, or are there not, more such segments between 0 and 10, than between 0 and 1? This is a yes/no question.

- JeffJo
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### Re: Cantor Diagonal Argument disproof

JeffJo wrote:Nobody (who understands math, and does not confuse the set-theory continuum with the space-time continuum

Just introducing a measure of reality even if most people find spacetime obscure.

) claims that points make up (I.e., provide the structure of) a line, like bricks make up a wall.

Good- even though the path of a point can define a curve.

But points do "lie on" the line.

They are arbitrarily imposed from the outside.

The property that a line segment it is infinitely divisible is what defines the continuum,

Here we go again with mathematicians redefining terms for their own purposes.

and provides the extra dimension.

The extra dimension is the line segment between end points.

But please tell me: are there, or are there not, more such segments between 0 and 10, than between 0 and 1? This is a yes/no question.

Loaded questions are "yes/no questions". An answer without an explanation is generally unacceptable. So, you can subdivide either interval with as many segments as you choose. For n segments there are n+1 endpoints. If you were to have infinite segments you would need infinite+1 endpoints, which is nonsensical. To have infinite endpoints you would get infinite-1 segments, which is also nonsensical. If you subdivide endlessly, you will tire yourself out eventually.

- robert 46
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### Re: Cantor Diagonal Argument disproof

One could make case for different "types" of infinities by constructing a line segment path which contains an infinity of points, and then by constructing a rectangle which contains an infinity of parallel line segments. Within the rectangle is therefore an infinity of points which 'exceeds' the infinity of points defining a line segment path.

As Robert points out (no pun intended) a line segment with N +1 interval end points contains N intervals- so the infinity of points and the infinity of intervals are 'different' in scale as well.

I suppose one could play the game 'which infinity is larger' based on all kinds of abstracted comparisons, but if infinity mean endless, then the practical benefit of such a mental exercise remains unclear to me.

As Robert points out (no pun intended) a line segment with N +1 interval end points contains N intervals- so the infinity of points and the infinity of intervals are 'different' in scale as well.

I suppose one could play the game 'which infinity is larger' based on all kinds of abstracted comparisons, but if infinity mean endless, then the practical benefit of such a mental exercise remains unclear to me.

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### Re: Cantor Diagonal Argument disproof

Edward Marcus wrote:One could make case for different "types" of infinities by constructing a line segment path which contains an infinity of points, and then by constructing a rectangle which contains an infinity of parallel line segments. Within the rectangle is therefore an infinity of points which 'exceeds' the infinity of points defining a line segment path.

Consequent to the situation that an infinite proper subset can have the same size (cardinality) of the parent set, the cardinality of points on a line is the same as on a plane, in a volume, in a hyper-volume, etc.

As Robert points out (no pun intended) a line segment with N +1 interval end points contains N intervals- so the infinity of points and the infinity of intervals are 'different' in scale as well.

It is better to think of the situation as endless points making endless intervals. Under no circumstances are there no intervals. However, consider the situation of a circle of shrinking radius. The area decreases, so what happens if the radius is reduced to zero? The area and circumference are also zero: this defines a point- it has position but no size. However, this is different from subdividing a line segment which must maintain its initial length. So whereas the circle can hypothetically be reduced to zero radius, we cannot consider the line segment to be partitioned into segments of zero length because the sum of infinite zeroes is zero.

I suppose one could play the game 'which infinity is larger' based on all kinds of abstracted comparisons, but if infinity mean endless, then the practical benefit of such a mental exercise remains unclear to me.

It means that when one is considering a fantasy world, it doesn't have to make sense.

- robert 46
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### Re: Cantor Diagonal Argument disproof

robert's non-answer here summarizes this entire issue:

The answer is "both yes and no." For any segment defined by 0<=X1<X2<=10, there is a corresponding segment defined by 0<=X1/10<X2/10<=1. So the number of such segments is the same. But for any segment defined by 0<=X1<X2<=1, there are nine more defined by N<=X1+N<X2+N<=N+1, where N is in {1,2,...,9}. There can even be fewer in the shorter interval. This is the property of "endless" that robert (and Gauss, Kronecker, Poincare, Weyl and others) refuse to address.

Mathematics is the science of producing a complete and consistent logical system from appropriately chosen definitions and axioms. Here, "appropriately" means that the ensuing deductions are complete and consistent. The fact that robert steadfastly refuses to provide his definitions, and to answer the above question, clearly shows that understands that he cannot do this.

I never said you couldn't try to explain your answer, but by not saying whether "yes" or "no" was an answer, robert intentionally evaded the issue. As he always does, and can't but do if he wants to perpetuate his fraud.robert 46 wrote:JeffJo wrote:But please tell me: are there, or are there not, more such segments between 0 and 10, than between 0 and 1? This is a yes/no question.

Loaded questions are "yes/no questions". An answer without an explanation is generally unacceptable.

The answer is "both yes and no." For any segment defined by 0<=X1<X2<=10, there is a corresponding segment defined by 0<=X1/10<X2/10<=1. So the number of such segments is the same. But for any segment defined by 0<=X1<X2<=1, there are nine more defined by N<=X1+N<X2+N<=N+1, where N is in {1,2,...,9}. There can even be fewer in the shorter interval. This is the property of "endless" that robert (and Gauss, Kronecker, Poincare, Weyl and others) refuse to address.

Points don't have paths.Good- even though the path of a point can define a curve.

It follows directly from facts robert has agreed to, that any line segment can be subdivided into two or more segments, and the endpoints of line segments are points. Misrepresentation like robert does here is how he thinks he can make his lies become truths.They are arbitrarily imposed from the outside.But points do "lie on" the line.

Here we go again with mathematicians redefining terms for their own purposes.

Mathematics is the science of producing a complete and consistent logical system from appropriately chosen definitions and axioms. Here, "appropriately" means that the ensuing deductions are complete and consistent. The fact that robert steadfastly refuses to provide his definitions, and to answer the above question, clearly shows that understands that he cannot do this.

It is only nonsensical in a system that does not accept, and define the properties of, infinite sets. Which is why they are inconsistent. Thank you for finally expressing that thought, and showing why a Mathematics based on "actually producing" every element t of such a set is "nonsensical".... To have infinite endpoints you would get infinite-1 segments, which is also nonsensical. If you subdivide endlessly, you will tire yourself out eventually.

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### Re: Cantor Diagonal Argument disproof

JeffJo wrote:Points don't have paths.robert 46 wrote:Good- even though the path of a point can define a curve.

They do with parametric equations such as the functions x(t), y(t), z(t) producing the position of a point in time.

It follows directly from facts robert has agreed to, that any line segment can be subdivided into two or more segments, and the endpoints of line segments are points.They are arbitrarily imposed from the outside.But points do "lie on" the line.

There is no disagreement that points can be put on a line.

[ The property that a line segment it is infinitely divisible is what defines the continuum,]

Here we go again with mathematicians redefining terms for their own purposes.

Mathematics is the science of producing a complete and consistent logical system from appropriately chosen definitions and axioms. Here, "appropriately" means that the ensuing deductions are complete and consistent.

Naïve set theory was loaded with contradictions. You can define a fantasy as rigorously as you like, but that doesn't mean it is sensible.

JeffJo wrote:I never said you couldn't try to explain your answer, but by not saying whether "yes" or "no" was an answer, robert intentionally evaded the issue.robert 46 wrote:JeffJo wrote:But please tell me: are there, or are there not, more such segments between 0 and 10, than between 0 and 1? This is a yes/no question.

Loaded questions are "yes/no questions". An answer without an explanation is generally unacceptable.

I was showing that JeffJo was setting up a trick question.

The answer is "both yes and no." For any segment defined by 0<=X1<X2<=10, there is a corresponding segment defined by 0<=X1/10<X2/10<=1. So the number of such segments is the same. But for any segment defined by 0<=X1<X2<=1, there are nine more defined by N<=X1+N<X2+N<=N+1, where N is in {1,2,...,9}. There can even be fewer in the shorter interval. This is the property of "endless" that robert (and Gauss, Kronecker, Poincare, Weyl and others) refuse to address.

Where an answer is "yes and no" there must be an equivocation.

It is only nonsensical in a system that does not accept, and define the properties of, infinite sets. Which is why they are inconsistent.[So, you can subdivide either interval with as many segments as you choose. For n segments there are n+1 endpoints. If you were to have infinite segments you would need infinite+1 endpoints, which is nonsensical.] ... To have infinite endpoints you would get infinite-1 segments, which is also nonsensical. If you subdivide endlessly, you will tire yourself out eventually.

So why does a system need to have infinite sets???

Thank you for finally expressing that thought,

Ambiguous as to which thought JeffJo is referring.

and showing why a Mathematics based on "actually producing" every element t of such a set is "nonsensical".

Recall that we are interested in producing an example, or producing an algorithm which would produce an example. ~D is not consequent to an algorithm; it is consequent to the definition of a bizarre infinite parallel access function- which is merely for the purpose of getting the answer some people want, but has no rational support.

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### Re: Cantor Diagonal Argument disproof

In physics, yes, the PARAMETRIC EQUATIONS define paths. Points themselves don't. And besides, this is Math Class.robert 46 wrote:JeffJo wrote:They do with parametric equations such as the functions x(t), y(t), z(t) producing the position of a point in time.

There certainly is. Points are not "put" anywhere. They ARE the endpoints of line segments.There is no disagreement that points can be put on a line.

Then it's a good thing that the Axiom of Infinity, and the Diagonalization Theorem, belong to Axiomatic Set Theory,, isn't it?Naïve set theory was loaded with contradictions. You can define a fantasy as rigorously as you like, but that doesn't mean it is sensible.

robert was, and still is, evading the question. Are there, or are there not, more line segments in [0,1] than in [0,10]? Explain all you want, but provide a yes or no with it.I was showing that JeffJo was setting up a trick question.

You mean, like you are doing now? To evade the question?Where an answer is "yes and no" there must be an equivocation.

Because when you don't have them, you get your nonsensical answers. Address the question, without equivocation, and you might see.So why does a system need to have infinite sets???

Only if you refuse to grasp the point that infinite segments, and "infinite+1" (whatever that is supposed to mean) endpoints is the nonsensical part.Thank you for finally expressing that thought,

Ambiguous as to which thought JeffJo is referring.

No, recall that what we are interested is defining an example, and that robert refuses to define what he thinks "producing" and "algorithm" mean, or why these undefined concepts are superior to a Mathematically valid algoritm that defines an infinite string.. He just insists that Cantor didn't do it, but won't say what was not done.and showing why a Mathematics based on "actually producing" every element t of such a set is "nonsensical".

Recall that we are interested in producing an example, or producing an algorithm which would produce an example.

The nth character of ~D is the opposite character of the nth character of the nth string in the mapping we assume exists. There is no part of ~D that is not "produced" by this well-defined algorithm; it is just a random-access algorithm instead of the sequential one robert seems to think is needed, but refuses to explain why.

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### Re: Cantor Diagonal Argument disproof

Which precisely is what cardinality is all about. It's a way of counting by pairing every member of a set to be counted with another set; if every member of each set finds a mate, the sets have the same "size" regardless of they being infinite or not.Edward Marcus wrote: One could make case for different "types" of infinities

Unfortunately, by the definition of cardinality, it can be shown that the set of points making up the unit square, for instance, has the same [1] cardinality as the unit line. However, despite that, there doesn't exist a continuous function from [0,1] to [0,1]*[0,1]. But there do exist s.c. "space filling curves", which are curves, i.e. one dimensional objects, that, in the limit, fill the whole unit square.Edward Marcus wrote:by constructing a line segment path which contains an infinity of points, and then by constructing a rectangle which contains an infinity of parallel line segments. Within the rectangle is therefore an infinity of points which 'exceeds' the infinity of points defining a line segment path.

The practical benefit is to have a system of comparing infinite sets - think pairing algorithms/mappings.Edward Marcus wrote:I suppose one could play the game 'which infinity is larger' based on all kinds of abstracted comparisons, but if infinity mean endless, then the practical benefit of such a mental exercise remains unclear to me.

Last edited by Gofer on Fri Jun 02, 2017 6:51 am, edited 1 time in total.

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### Re: Cantor Diagonal Argument disproof

Robert's idea of "producing" is to actually calculate all the elements of ~D, which, however, there's no need for, because we only need to prove that ~D can't equal any row. Here's an analogy and exercise for Robert:

Suppose that x=5+s, where s=1+2+3+4+5; do we really need to calculate all of s to prove that x>5?

Suppose that x=5+s, where s=1+2+3+4+5; do we really need to calculate all of s to prove that x>5?

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